#499566

order	512
nilpotency class	4
isoclinism	isoclinic class of #499054
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#481
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂⁴	C₂×SD₁₆
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

552 conjugacy classes of subgroups; only the 218 normal subgroups are shown.

normal subgroups only

This group has 552 conjugacy classes of subgroups — too many to draw. Only its 218 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 40 points:

(1,2,5,7)(3,8,4,6)(9,10)(11,15)(12,18)(13,27)(14,19)(16,26)(17,30)(20,28)(21,38)(22,33)(23,40)(24,32)(25,29)(31,35)(34,39)(36,37);(1,3,5,4)(2,6,7,8)(9,11,12,21,14,22,26,24)(10,15,18,38,19,33,16,32)(13,28,25,23,30,31,36,39)(17,35,37,34,27,20,29,40);(1,4,5,3)(2,6,7,8)(9,10)(11,20)(12,16)(13,29)(14,19)(15,31)(17,36)(18,26)(21,34)(22,35)(23,38)(24,40)(25,27)(28,33)(30,37)(32,39);(9,12,14,26)(10,16,19,18)(11,21,22,24)(13,25,30,36)(15,32,33,38)(17,29,27,37)(20,34,35,40)(23,31,39,28)

Polycyclic presentation:

2^9|a^2=e,b^2=efh,c^2=e,d^2=i,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=di,g^a=gi,c^b=cg,f^b=fi,g^b=gh,h^b=hi,g^c=ghi,h^c=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 406762);